Before getting to the main topic of today's post, I have a couple of announcements to make!

First, I've normally tried to post on Thursdays and Sundays on this blog, but regular commitments are taking priority, so I'm going to be posting on Sundays only for the time being.

Second, I recently created a new subreddit all about mental math techniques. If you're on reddit.com, please subscribe and contribute your own thoughts and links! Since it's often quicker to post links there, this may hopefully compensate for the difference in the blog posting schedule.

The rest of today's post will be about calculating powers of e in your head. Wait...what exactly is e?!? That's a great place to start.

BASICS: When mathematicians refer to e, they're referring to a specific number, approximately 2.71828 (it goes on forever, just like Pi), that is used to calculate continuous growth.

What exactly will you be working out in your head? Given an exponent x, you'll be able to take a problem of the form ex, and turn it into a power of 10 (in other words, you'll figure out y in 10y).

The technique I'll teach is based on the fact that we can take the general problem ex = 10y and isolate y by turning the problem into a logarithm: log10(ex) = y. Thanks to standard logarithm rules, we can turn this into the following form: x × log10(e) = y. A little help from Wolfram|Alpha tells us that this can further be simplified to the following form: x × 0.4342944... = y

OK, I hear you say, we've turned a seemingly complex exponential problem into a multiplication problem, but it doesn't seem easy to multiply by a number like 0.4342944 in your head! What we're actually going to be doing is approximating this number by multiplying by 0.4343. Yes, we'll be adjusting for the approximation, as well. Further, we're going to break that up into several steps, by turning it into the form (43 × 101)⁄10,000.

Yes, I have a few tricks up my sleeve to help you make these calculations at each stage, and I'll teach them in the next section.

TECHNIQUE: Let's start with a simple example of working out e16 as we go through each of the following steps.

MULTIPLYING BY 43: Take the exponent and multiply it by 4. In our example, 16 is the exponent, and you should be able to quickly work out that 16 × 4 = 64.

Next, take the exponent and multiply it by 3. Continuing with our example, 16 × 3 gives us a total of 48.

To make things easier, we need to split this 2nd answer up into the 1s digit and everything else. In our example, instead of thinking of the answer as simply 48, think of it as + 4 with an 8 slapped on the end. You'll always think of the multiple of 3 in this way. If multiplying by 3 resulted in, say, 312, you'd think of that as + 31 with a 2 slapped on the end. This type of thinking will make it easier for the next step.

To complete this step, add the two answers together. In our example, we'd add 64 + 4 with an 8 slapped on the end. The answer then, is 64 + 4, which is 68, with an 8 slapped on the end, which is 688. Yes, 43 × 16 is 688.

Dealing with say, 23? 23 × 4 = 92 and 23 × 3 = + 6 with a 9 slapped on the end. 92 + 6 with a 9 slapped on the end is equal to 98 with a 9 slapped on the end, or 989. Practice multiplying by 43 in this way before moving on. You may be surprised how quickly you get used to it.

For this feat, however, you'll have to be able to deal with multiplying 3- and 4-digit numbers by 101. Don't worry, though, it's still simple.

To multiply a 3- or 4-digit number by 101, split the original number up, with the 10s and 1s digit off to the right, the remaining digits off to the left, and imagine two placeholders in-between them.

So, continuing with our example, 688 would be split up into 6__88. That's 6, followed by 2 placeholders, followed by 88, the original 10s and 1s digit.

To fill the placeholders, simply add the numbers on either side of the placeholders. 6 + 88 = 94, so we put 94 in those empty spaces, resulting in the number 69488. Yes, you've just mentally worked out the answer to 16 * 4343 in your head!

You do have to watch for the times when the digits on either side of the place holder add up to more than 100. In that case, you have to add the 1 to the digits on the left to compensate.

For example, if you multiplied 44 by 43 earlier and worked it out to be 1892, you'd then split that into 18__92, and then add 18 + 92 to get 110. The 10 part goes in the middle, and the 1 from the hundreds place is added to the 18 to get 19. The result is 191092. This is simple, once you get used to it.

DIVIDING BY 10,000: This is the simplest part of all! Just take the number from the previous step, and move the decimal 4 places to the left.

In our continuing example, we worked out the number 69488. Moving the decimal 4 places to the left turns the number into 6.9488!

To make your result more accurate, simply drop the last (rightmost) digit of your answer. Don't round up or down, just completely ignore the last digit!

For our example result of 6.9488, we'd drop the rightmost digit to get 6.948. With that step, you're done. In your head, you worked out that e16 ≈ 106.948. How close is that to the actual solution? According to Wolfram|Alpha, the actual answer is 106.9487, so your mental estimate is accurate as far as it goes!

Naturally, if the rightmost digit of your calculation is 0, this is done for you almost automatically.

THE FULL PROCESS ALL AT ONCE: To make sure you've got this, let's try and work out a bigger problem. This time, we'll calculate e27 = 10y.

27 × 4 = 108 and 27 × 3 = +8 with a 1 slapped on the end.

108 + 8 with a 1 slapped on the end is equal to 116 with a 1 slapped on the end, or 1161.

• By looking at the whole number part of the answer (the significand) and adding 1 to that, you can state the number of digits the full answer would have. In our 6.9488 example, we take the whole number part, the 6, and add 1 to get 7, so we can state that the answer is a 7-digit number. Having worked out e135 to be about 1058.63, we can state that the answer is a 59-digit (58 + 1) number!

• To prevent yourself from having to deal with multiplying 5-digit numbers from 101, simply avoid exponents greater than 232. Being able to estimate up to e232 in your head is still impressive!

• You can handle exponents with a decimal in them by working them out as if they were a whole number, and then adjusting for an appropriate number of additional decimal points when you're moving the decimal point. For example, e1.6 is the same as e16, but with the decimal moved one place more to the left. Since e16 ≈ 106.948, it's easy to see that e1.6 ≈ 100.6948.

This time around, we've got a round up of math designed to amaze and surprise you!

• @LucasVB is the designer behind some of the most amazing math-related graphics I've ever seen. You can see some of his amazing work at his tumblr site, and even more at his Wikimedia Commons gallery. Even if you don't understand the mathematics or physics behind any given diagram, they're still enjoyable, and may even prompt your curiosity.

• Our old friend @CardColm is back with more math-based playing-card sneakiness! In his newest Postage Stamp Issue post, he presents a sneaky puzzle that you can almost always win. After shuffling cards, the challenge is to cut off a portion of cards, and see how many of the numbers from 1 to 30 you can make using just the values of those cards. It seems very fair and above-board, but the math behind it allows you to win almost every time!

I do have a few criticisms of that video. First, I think they should've gone a little further, and shown a complete multiplication of a 3 by 2 and a 2 by 3 matrix, just for completeness sake. Second, did anyone else get confused by the mention of linear algebra? Doesn't it seem strange that a rectangular arrangement of numbers should be so important in something that begins with the word linear?

Once again, BetterExplained.com's Kalid Azar comes to the rescue with An Intuitive Guide to Linear Algebra. He takes things quite a bit slower than the video, but he does this so you can grasp all the concepts.

Kalid starts, thankfully, with an explanation of just what is so linear about matrices and linear algebra. He then shows you how to get your head around the unusual notation and multiplication. His analogy to mini-spreadsheets is a very helpful!

Probably the biggest help is the real-world example, instead of the classic vectors that are often used. He even briefly covers scary terms such as eigenvector.

Do take the time to go through these resources, as the matrix is a fascinating tool that shouldn't be ignored.

Welcome to a geeky, yet still special, day of the year that is close to my heart!

Vi Hart kicked things off with an anti-Pi rant. This might seem like a strange action to take on Pi Day, but she does it in her own inimitable style, and really does make some good points about Pi in the process:

As many regular Grey Matters readers realize, I love it when something fun like this can lead to a better understanding of the math behind it.

According to the above video, we can also expect a more detailed video about the Euler characteristic mentioned in this video. You can also find out more about Brussels Sprouts and other dot games in Martin Gardner's books Mathematical Carnival and The Colossal Book of Mathematics.

Have fun and play around with these games. You never know what else you may learn!

Trigonometry is a subject that can strike fear into the heart of almost any high school student.

It's actually quite understandable and useful if taught clearly enough so you grasp it. In this post, I'll show you just where to find those resources.

BASICS: As with any topics, you'll want to make sure you have a few basics down. When getting started with trigonometry, that means being clear on the concepts of similarity, and the good old Pythagorean Theorem.

The series Project Mathematics! is of great help here. Their amazing computer-animated lessons make the concepts of similarity clear and even simple:

What makes this post so great? He slowly introduces each concept and makes each step concrete and understandable. Kalid starts by likening trigonometry to learning anatomy, as if you were learning the anatomy of a circle. After his introduction to sine and cosine with a dome analogy, he points out that the numbers you're seeing are percentages. How much bigger or smaller is this part or that part compared to the radius of the circle? Cleverly, he even goes back to his anatomy metaphor to make this more understandable.

With the idea of a wall next to the dome, he then introduces the tangent and the secant, and then uses a ceiling built over the dome to help drive home the ideas of the cotangent and the cosecant. Probably the most startling moment in the whole post, however, is when Kalid gets you to see the connections between the 3 types of triangles he's been explaining. When you see that they're simply scaled versions of each other, everything begins to fall in place!

Once you have the scaled triangles in your mind, your knowledge of the Pythagorean Theorem and similar triangles make the relationships almost trivial to work out in your head! Instead of memorizing formulas you'll quickly forget after a trig test, you simply grasp the relationships, and can work them out anytime you need them!

Even as good as Kalid's explanations are, he points out that you shouldn't get too attached to the static diagrams. Taking that advice, I used the online graphing calculator as Desmos.com to create some models I could play around with to grasp the concepts for myself, and I've linked to them using the corresponding section names from Kalid's article:

I sincerely hope you take the time to explore most, if not all, of these resources, as they gave me a new respect and understanding for the tools of trigonometry, and I simply want to share that joy of discovery.